Image Denoising based on Sparse Representation and Dual Dictionary

Transcription

1 ISSN: , Volume-4 Issue-4, April 015 Image Denoising based on Sparse Representation and Dual Dictionary Anees G Nat, Sreeram G, Sarafudeen K, Sreeraj M C Abstract earning-based image denoising aims to reconstruct a denoised image from te prior model trained by a set of noised image patces. In tis paper, we address te image denoising problem, were zero-mean wite and omogeneous Gaussian additive noise is to be removed from a given image. Image denoising metod via dual-dictionary learning and sparse representation consists of te main dictionary learning and te residual dictionary learning to recover denoised image. Te approac taen is based on sparse representations over trained dictionaries. Using te K-SVD algoritm, we obtain a dictionary tat describes te image content effectively. Using te corrupted or noised image primary main dictionary training is done. Since te K-SVD is limited in andling small image patces, we extend its deployment to arbitrary image sizes by defining a global image prior tat forces sparsity over patces in every location in te image. We provide a residual dictionary learning pase wic leads to a simple and effective denoising mecanism. Tis leads to a better denoising performance, and surpassing recently publised leading alternative denoising metods. xtensive experimental results on test images validate tat by employing te proposed two-layer progressive sceme, more image details can be recovered and muc better results can be acieved in terms of bot and visual perception. Index Terms sparse representation, dictionary learning, image denoising, K-SVD, residual dictionary. I. INTRODUCTION Image denoising is critical problem tat as been implemented by a variety of tecniques tat rely on implicit and explicit modeling of noise, wic is often assumed to ave a Gaussian distribution function. Te denoising approaces tat ave been developed can broadly be classified into local and nonlocal metods. Wile local metods uses weigted averaging and ave a limited estimation support, wile nonlocal metods utilize redundancy and wor on larger support regions to provide better statistics. Image denoising deals wit te problem of reconstructing an image x from its measurements y, measured in te presence of an additive zero-mean wite and omogeneous Gaussian noise v, wit standard deviation σ. Te measured image is, tus y = x + v (1) Ten te problem is to design an algoritm wic removes te noise from y, maing a close reconstruction of original image x possible. Te linear relationsips among ig-dimension signals are accurately recovered from teir low-dimension Manuscript Received on April 015. Anees G Nat, Department of CS, TKM College of ngg.,kollam, Kerala, India. Sreeram G, Student, Department of CS, TKM College of Sarafudeen K, Student, Department of CS, TKM College of Sreeraj M C Student, Department of CS, TKM College of projections by sparse representation. Instead of woring directly wit te patc pairs sampled from ig- and low-resolution images, based on te assumption tat for a given noised patc, te sparse representation vector in te over-complete dictionary trained by noised images is te same as te one of its corresponding noiseless patc in te over-complete dictionary trained by ig-resolution images, leant a compact representation for tese patc pairs to capture te co-occurrence prior to improve te speed and te robustness significantly, acieving state-of-te-art performance. ately, embaring from te algoritm, a modified approac is proposed, wic sows to be more efficient and muc faster. However, due to te restriction of te over-complete dictionary s size and te intrinsic sparsity of te algoritm, one limitation in recovering ig-frequency details sould be noticed. It s difficult to recover te details of te noiseless image in te corresponding original image completely from te initial interpolation of an input noised image, for te gap between te frequency spectrum of te corresponding original image and tat of te initial interpolation is so wide tat learning-based algoritm usually can t wor well. In tis paper, te noiseless image to be estimated is considered as a combination of two components: main ig-frequency (MHF) and residual ig-frequency (RHF), and we develop a novel metod for learning-based image denoising via sparse representation, wic consists of dual-dictionary learning levels: main dictionary learning and residual dictionary learning, corresponding to recovering MHF and RHF, respectively. Troug te proposed two-layer algoritm, in wic details of ig-frequency are estimated by a progressive way, te main ig-frequency is first recovered to reduce te gap of te frequency spectrum primarily, and ten te residual ig-frequency is reconstructed to enance te result effectively wit a sorter gap of te frequency spectrum. II. PRVIOUS WORKS IN IMAG DING For last five decades tis problem as been addressed in diverse points of view. Statistical estimators of all sorts, spatial adaptive filters, stocastic analysis, partial differential equations, transform-domain metods, splines and oter approximation teory metods, morpological analysis, order statistics, and more, are some of te many directions explored in studying tis problem. Denoising based on sparsity became an interested researc area in te past decade. At first, sparsity of te unitary wavelet coefficients was considered, leading to te srinage algoritm [1] [9]. One reason to turn to redundant representations was te desire to ave te sift invariance property [10]. Also, wit te growing realization tat regular separable 1-D wavelets are inappropriate for andling images, several new tailored multiscale and directional redundant transforms were introduced, including te curvelet [11], [1], contourlet [13], 49

2 Image Denoising based on Sparse Representation and Dual Dictionary [14], wedgelet [15], bandlet [16], [17], and te steerable wavelet [18], [19]. In parallel, te introduction of te matcing pursuit [0], [1] and te basiss pursuit denoising [] gave rise to te ability to address te image denoising problem as a direct sparse decomposition tecnique over redundant dictionaries. III. PROPOSD DUA RY BASD SCHM Te proposed sceme consists of a dictionary learning stage tat trains dual dictionaries[6], namely main dictionary (MD) and residual dictionary (RD), and an image reconstruction stage, performing te image denoising on te input image using te trained model from te previous stage. A preliminary denoising is applied on te noised image to mae it easier to apply to dictionary learning [3],[4] pase. A. Dictionary earning: In tis stage, two dictionaries are trained using sparse representation, i.e. MD and RD, wic correspond to te recovery of MHF and RHF, respectively. Many metods ave been proposed for dictionary learning by sparse representation. Here, we adopt te K-SVD[5]. Te dictionary learning stage starts by collecting a set of training images witout any noise. A training images witout any noise denoted by H, at first noise is added to te training image ( H ) and down sampled to yield a corresponding noisy low-frequency image wic is ten filtered by using a preliminary filter, a noisy low-frequency image is constructed, denoted by H in Fig wic is of te same size as H. Ten, real noisy ig-frequency image H is generated by subtracting H from H. Fig. 1 Illustration of dictionary learning stage. Ten, MD will be built, wic is actually a combination of two coupled sub-dictionaries: low-frequency main dictionary (MD) and ig-frequency main dictionary (HMD). Wit H and H, local patces are extracted forming te training data TD = p,p. p is te set of patces extracted from te ig-resolution image H directly, and p mean tose patces built by first extracting patces from filtered images obtained by filtering H wit certain ig-pass filters suc as aplacian ig-pass filters, and ten reducing te dimensions by Principal Component Analysis (PCA) algoritm. Next, te K-SVD dictionary training is applied to te set of patces p generating MD: MD,{ q } = arg min MD,{ q } Were are sparse representation vectors, and. is te norm counting te nonzero entries of a vector. Based on te assumption tat te patc! can be recovered by approximation as HMD. HMD can be defined by minimizing te following mean approximation error, i.e., HMD = arg min Σ p HMD HMD = arg min Σ p Were te matrices and Q consist of p, and, respectively. Terefore, te solution can be solved as follows (given tat Q as full row ran) ): ' ( ' ) *+ (5) Finally, te residual dictionary will be trained in te following steps. Wit te main dictionary and H te noisy main ig-frequency image denoted by H is produced by virtue of image reconstruction metod wic will be introduced in te next subsection. Utilizing H, te noisy temporary image denoted by H wic contains more details tan H and te noisy residual ig-frequency image denoted by H are generated, as sown in Fig. Tus, RD can be built wit te input of H and H using te same dictionary learning metod as MD. It is important to note tat RD also consists of two coupled sub-dictionaries: low-frequency residual dictionary (RD) and ig-frequency residual dictionary (HRD), and bot MD and RD mae up te dual-dictionaries. A. Sparsity based Dictionary creation Creation of bot MD and RD is done by two steps. ow resolution dictionary training and ig resolution dictionary training. Te procedure for training dictionaries is adopted from [4] s metod. ow Resolution Dictionary Training: Te dictionary training stage starts wit te low-resolution patces! # $ extracted from low resolution image. As te result of applying K-SVD dictionary learning algoritms to tese patces, te dictionary, - / 01 is created. Tis training process also generates te sparse representation coefficient vectors corresponding to te training patces! # $ as a side product. Here is te l 0 norm gives te count of nonzeroo values in te vector. Hig Resolution Dictionary earning: By approximating p H q, te HR patc p can Σ pl MD. q s. t q K () HMD. q (3) HMD.Q (4) D 50

3 ISSN: , Volume-4 Volume Issue-4, April 015 Te OMP algoritm is applied to generate p and te sparse representation vectors is built by allocating 5 atoms to teir representation. Te representation 5 vectors are multiplied by HMD, reconstructing ig-resolution patces by!. = (6) Defining / te operator wic extracts a patc from te ig resolution image in location. Te main ig-frequency image denoted by H is generated by solving te following minimization problem: H MHF = arg min p HMD.Q H MHF (7) Wic as a closed-form form east-square east solution, given by H MHF = [ RT R ] 1 RT pˆ (8) Fig.. Single ingle dictionary learning in detail be recovered. For tis te already generated sparse representation vector of R patc ig-resolution dictionary dictionary q is multiplied wit te H D. Te ig resolution resolut H D can be found ound to get te correct approximation. So, H D is te dictionary matrix wic minimizes te approximation error. B. Image Reconstruction Image reconstruction stage attempts to magnify an input noisy image, wic is assumed to be generated g from an original image by adding te same noise as used in te above learning stage. Te final estimated image is reconstructed by using dual dictionaries successively and more ig-frequency ig details are added progressively, as illustrated in Fig 3. To begin wit, an input noisy image to produce a noisy low-frequency image denoted by H. Combining H and MD, te main ig-frequency frequency image denoted by H is generated employing te image reconstruction metod using Sparse-Representations. Fig. 3. Illustration of image reconstruction stage. Concretely, H is filtered wit te same ig-pass ig filters and PCA projection as te training stage, and ten is decomposed into overlapped patces p ingle dictionary reconstruction in detail Fig.4. Single IV. RSUTS Te proposed algoritm is implemented in MATAB R01a using K-SVD SVD for learning dictionary and OMP algoritm for sparse approximation. Te computer system used is wit Intel Core i5 410M CPU at.30ghz wit 4GB of RAM. Set of sample images im are collected and zero-mean mean wite and omogeneous Gaussian additive noise as been added wit different variances var ranging from to 0.1. In dictionary training stage adding more and more images for training would lead to improved results. Te feature extraction from te low resolution image is performed wit four filters: f1 = [1, 1] = ft and f3 = [1,, 1] = f4t. Te size of eac patc is set to n = 81 (9 9), and after aft applying PCA te dimensionality as been reduced from 34 (4 81) dimensions to 30 dimensions. In dictionary training, number of iterations in K-SVD SVD algoritm is set to 40, wit number of atoms in dictionary as m = Te value of wic defines tee sparsity is given as 3. For evaluating te performance of te proposed system wit varying amount of noises,, we too noise variances ranging from to 0.1 for standard 3 images of varying sizes. 51

4 Image Denoising based on Sparse Representation and Dual Dictionary Analyzing te and SSIM comparison of te results wic is also sown as graps, it is clear tat using two levels of dictionary provides better and visual qualities. Table 1: &SSIM for various noise levels for ena.tif VARIANC Y IMAG RY +RSIDUA RY +RSI DUA DICTI ONAR Y SSIM Fig. 6. comparison grap of Peppers.tif Table 3: &SSIM for various noise levels for HR-067.tif VARIAN C Y IMAG RY +RSIDUA RY +RSIDUA RY SSIM Fig. 5. comparison grap of ena.tif Table : &SSIM for various noise levels for peppers.tif +RSID Y VARIANC UA IMAG +RSIDUA DICTIO RY NARY RY SSIM Fig. 7. comparison grap of HR-067.tif 5

5 ISSN: , Volume-4 Issue-4, April 015 V. CONCUSION Tis wor presents an improved image denoising approac for creating noiseless images from noisy samples using two dictionaries. Tis wor as presented a simple metod for image denoising, leading to state-of-te-art performance, equivalent to and sometimes surpassing recently publised leading alternatives. Te proposed metod is based on local operations and involves sparse decompositions of eac image bloc under one fixed over-complete dictionary, and a simple average calculations. Te content of te dictionary is of prime importance for te denoising process we ave sown tat a dictionary trained for natural real images, as well as an adaptive dictionary trained on patces of te noisy image itself, bot perform very well. RFRNCS [1] D.. Donoo and I. M. Jonstone, Ideal spatial adaptation by wavelet srinage, Biometria, vol. 81, no. 3, pp , Sep [] D.. Donoo, De-noising by soft tresolding, I Trans. Inf. Teory, vol. 41, no. 3, pp , May [3] D.. Donoo, I. 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Ma, Image super-resolution as sparse representation of raw image patces, Proceedings of I Conference on Computer Vision and Pattern Recognition, pp. 1-8, 008. [4] R. Zeyde, M. lad, and M. Protter, On Single Image Scale-Up using Sparse-Representations, Curves & Surfaces, NCS 690, pp , 011. [5] M. Aaron, M. lad, and A. Brucstein, K-SVD: An algoritm for designing overcomplete dictionaries for sparse representation, I Transactions on Signal Processing, vol. 54, no. 11, pp , Nov [6] Jian Zang, Cen Zao, Ruiqin Xiong, Siwei Ma, Debin Zao, Image Super-Resolution via Dual-Dictionary earning And Sparse Representation, I Int. Symposium of Circuits and Systems (ISCAS) 01 Anees G Nat, personal profile wic contains teir education details, teir publications, researc wor, membersip, acievements, wit poto tat will be maximum words. Sreeram G, personal profile wic contains teir education details, teir publications, researc wor, membersip, acievements, wit poto tat will be maximum words. Sarafudeen K, personal profile wic contains teir education details, teir publications, researc wor, membersip, acievements, wit poto tat will be maximum words. 53